40 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA
From the definition of d, it follows immediately that d is left-invariant
(because g is), satisfies the triangle inequality, is symmetric and positive.
Moreover, the right-hand inequality in (12.9.1.2) is obvious, and shows that
d(xy x) = 0 for all x e G. Hence dis & pseudo-distance on G. To prove the left-
hand inequality of (12.9.1.2) we shall show by induction on p that, for each
finite sequence (z£)0^i^p of p 4- 1 points in G such that z0 = x and zp = y, we
have
(12.9.13) I W W > iflfo JO-
£ = 0
P- 1
This inequality is obvious if /? = !. Let us write a = 2_,g(zi9 zt+i)- The
i = 0
inequality (12.9.1.3) is true if a ^ %, because y(x9y)^l. If a = 0, then
7. = z.+1 for 0 ^ f :g/> — 1, hence ;c = y and so (12.9.1.3) is trivially satis-
fied. So suppose that 0 < a < ^, and let /z be the greatest index such that
£ 0(zi9 zi+1) £ ia. Then we have % g(zt,zi+1) > -Jot, so that
By the inductive hypothesis, we have g(x, zh) ^ a and g(zh + i, y) ^ a; on the
other hand, it is clear that g(zh,zh+i) ^ a. Let k be the smallest integer >0
such that 2~* ^ a. Then A: ^ 2, and the elements x~1zh , z^lzh+1, z^^y are all
in Vt , by virtue of the definition of g. Hence x"1y e V% c= Vfc^.1? which implies
that ^(x, j) ^ 21"* ^ 2a5 and proves (12.9.13).
Hence (12.9.1.2) is established, and shows first of all that dis a distance on
G. Also, if r > 0, the ball &'(e; r) (with respect to d) contains V* for all indices
such that 2~"k<r, and conversely each Vk contains the ball B'(e; 2~*~~1).
Since dis left-invariant, it now follows from (12.8.3(i)) and (12.2.1) that d
defines the topology of G.
In general it is not possible to define the topology of G by means of a
distance which is both left- and right-invariant (Problem 1 and Section 14.3,
Problem 11).
(12.9.2) Let G be a metrizable group. A necessary and sufficient condition
for a sequence (xj in G to be a Cauchy sequence with respect to a left-invariant
distance defining the topology ofG is that, for each neighborhood V ofe, there
exists an integer n0 such that x~lxm e Vfor all m ^ «0 and n^.n0.
For if d is a left-invariant distance, we have
d(xn,*^ = d(e,x~lxm\
and the proposition follows from the fact that d defines the topology of G.duct of the topological groups G« ,